Let P0 = $1000.00.
• k = 0.5% = _____0.005______ as a decimal.
Compute and explain how to find the doubling time. Then, write the compound interest equation for this value of k and the 3 values for t.
Number of years Equation New Value
1 P(1)=$1000e^(0.005)(1)
5 P(5)=$1000e^(0.005)(5)
10 P(10)=$1000e^(0.005)10)
To find the doubling time, we need to use the formula for compound interest, which is given by:
A = P(1 + r)^t
where:
A is the final amount
P is the initial amount (principal)
r is the interest rate (as a decimal)
t is the number of time periods
In this case, our initial amount (P0) is $1000.00 and the interest rate (k) is 0.5% (as a decimal, it becomes 0.005). We want to find the value of t when the final amount (A) is double the initial amount (2P0).
So let's substitute the values into the formula and solve for t:
2P0 = P0(1 + 0.005)^t
Divide both sides by P0:
2 = (1 + 0.005)^t
Now we need to solve for t. Taking the natural logarithm (ln) of both sides:
ln(2) = ln((1 + 0.005)^t)
Using the property of logarithms, we can bring down the exponent:
ln(2) = t ln(1 + 0.005)
Finally, divide both sides by ln(1 + 0.005) to solve for t:
t = ln(2) / ln(1 + 0.005)
Now we can calculate the doubling time by evaluating this expression using a calculator or computer software.
Once we find the value of t, we can use it to calculate the compound interest at different time periods.
For 1 year:
P(1) = P0 * e^(k * t)
P(1) = $1000 * e^(0.005 * 1)
For 5 years:
P(5) = P0 * e^(k * t)
P(5) = $1000 * e^(0.005 * 5)
For 10 years:
P(10) = P0 * e^(k * t)
P(10) = $1000 * e^(0.005 * 10)
By substituting the respective values of t into these equations, we can calculate the new values.